Can you choose the order that multiple triggered abilities from the same event go on the stack?

When multiple triggered abilities trigger at the same time, they are placed on the stack in a specific order determined by who controls them. If you control two or more triggered abilities that trigger from the same event, you choose the order in which they're placed on the stack (CR 603.3b). Because the stack is last-in, first-out, the ability you place on the stack last will resolve first.

If both you and an opponent each control triggered abilities from the same event, the Active Player (the player whose turn it is) puts all of their triggered abilities on the stack first, then each other player in turn order puts theirs on the stack (CR 603.3b). This means the non-active player's abilities will resolve first overall, since they were added to the stack last.

Each player only chooses the relative order of their own triggered abilities — you cannot interleave your triggers with your opponent's triggers in a custom order.

Concrete example: You control both Purphoros, God of the Forge and Warstorm Surge, and you play a creature. Both abilities trigger simultaneously. You choose to put Purphoros's trigger on the stack first, then Warstorm Surge's trigger on top. Warstorm Surge resolves first (dealing damage equal to the creature's power), then Purphoros resolves (dealing 2 damage to each opponent).

This ordering choice can be strategically important, so always consider which ability you need to resolve first before announcing the stack order.